CS 70 Lecture 19 - CS 70 Spring 2010 Alistair Sinclair...

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CS 70 Discrete Mathematics and Probability Theory Spring 2010 Alistair Sinclair Note 19 A Brief Introduction to Continuous Probability Up to now we have focused exclusively on discrete probability spaces , where the number of sample points ω is either Fnite or countably inFnite (such as the integers). As a consequence we have only been able to talk about discrete random variables, which take on only a Fnite (or countably inFnite) number of values. But in real life many quantities that we wish to model probabilistically are real-valued ; examples include the position of a particle in a box, the time at which an certain incident happens, or the direction of travel of a meteorite. In this lecture, we discuss how to extend the concepts we’ve seen in the discrete setting to this continuous setting. As we shall see, everything translates in a natural way once we have set up the right framework. The framework involves some elementary calculus but (at this level of discussion) nothing too scary. Continuous uniform probability spaces Suppose we spin a “wheel of fortune” and record the position of the pointer on the outer circumference of the wheel. Assuming that the circumference is of length ° and that the wheel is unbiased, the position is presumably equally likely to take on any value in the real interval [ 0 ] . How do we model this experiment using a probability space? Consider for a moment the (almost) analogous discrete setting, where the pointer can stop only at a Fnite number m of positions distributed evenly around the wheel. (If m is very large, then presumably this is in some sense similar to the continuous setting.) Then we would model this situation using the discrete sample space = { 0 , ° m , 2 ° m ,..., ( m 1 ) ° m } , with uniform probabilities Pr [ ω ]= 1 m for each ω . In the continuous world, however, we get into trouble if we try the same approach. If we let ω range over all real numbers in [ 0 ] , what value should we assign to each Pr [ ω ] ? By uniformity this probability should be the same for all ω , but then if we assign to it any positive value, the sum of all probabilities Pr [ ω ] for ω will be ! Thus Pr [ ω ] must be zero for all ω . But if all of our sample points have probability zero, then we are unable to assign meaningful probabilities to any events! To rescue this situation, consider instead any non-empty interval [ a , b ] [ 0 ] . Can we assign a non-zero probability value to this interval? Since the total probability assigned to [ 0 ] must be 1, and since we want our probability to be uniform, the logical value for the probability of interval [ a , b ] is length of [ a , b ] length of [ 0 ] = b a ° . In other words, the probability of an interval is proportional to its length. Note that intervals are subsets of the sample space and are therefore events . So in continuous probability, we are assigning probabilities to certain basic events, in contrast to discrete probability, where we assigned probability to points in the sample space. But what about probabilities of other events? Actually, by speci-
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This note was uploaded on 09/21/2010 for the course CS 70 taught by Professor Papadimitrou during the Spring '08 term at University of California, Berkeley.

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CS 70 Lecture 19 - CS 70 Spring 2010 Alistair Sinclair...

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